CN102759734A - Imaging method with improved range migration algorithm (RMA) for high-resolution spaceborne synthetic aperture radar (SAR) - Google Patents

Abstract

In order to overcome the defect that a traditional range migration algorithm (RMA) cannot be directly applied to echo data processing of a high-resolution spaceborne synthetic aperture radar (SAR), the invention provides an imaging method with an improved RMA for the high-resolution spaceborne SAR, and belongs to the technical field of SAR imaging. Aiming at the problem of the high-resolution spaceborne SAR, the imaging method adopts a new slant range model to accurately model a curved track of the high-resolution spaceborne SAR; a two-dimensional frequency spectrum of echoes of the high-resolution spaceborne SAR is derived on the basis of the slant range model; and range migration correction, stolt interpolation and azimuth focusing of the traditional RMA are improved. By adopting the improved imaging algorithm, imaging process of the echoes of the high-resolution spaceborne SAR can be effectively completed.

Description

A kind of formation method that is used for the improvement range migration correction algorithm of High Resolution Spaceborne SAR

Technical field

The invention belongs to the synthetic aperture radar image-forming technical field, relate to the formation method that a kind of High Resolution Spaceborne SAR is improved the range migration correction algorithm.

Background technology

Satellite-borne synthetic aperture radar (SAR) is a kind of modern high-resolution microwave side-looking imaging radar of round-the-clock, round-the-clock; It fully utilizes synthetic aperture technique, pulse compression technique and data processing technique; Adopting short antenna just can obtain higher range resolution and azimuthal resolution, is a kind of advanced person's of current microwave regime telemetry over the ground.Along with the continuous development of Spaceborne SAR System and perfect; At mapping, crops the yield by estimation, ocean and hydrologic observation, environment and disaster monitoring, resource exploration; And scientific domain such as military surveillance demonstrates powerful advantage day by day, aspect earth observation, bringing into play more and more important effect.Use so widely for satellite-borne SAR, its application quality quality generally depends on the image resolution ratio of satellite-borne SAR.Through the development of many decades, existing Spaceborne SAR System has reached about 1m such as the resolution of TerraSAR-X and Cosmo-Skymed.Predictably, in future soon, the satellite-borne SAR of sub-meter grade resolution will launch.

High-resolution Spaceborne SAR System needs high-resolution imaging algorithm.In traditional imaging algorithm; Range migration correction algorithm RMA (Range Migration Algorithm) algorithm is more accurate imaging algorithm; The RMA algorithm transforms to two-dimensional frequency to FFT and orientation to FFT through distance earlier; Accomplish distance to focusing in two-dimensional frequency; Accomplish range migration correction and orientation to focusing on (distance is accomplished to focusing on a lasted step, does not need orientation migration correction) through the stolt interpolation then, again through apart to IFFT and the orientation image after IFFT obtains focusing on.Yet traditional RMA algorithm can not directly apply to the echo data of High Resolution Spaceborne SAR and handle, because traditional RMA algorithm thinks that track is a straight line, and the relative velocity of satellite and target does not change with oblique distance.

Summary of the invention

The present invention is directed to traditional RMA algorithm and can not directly apply to the defective that the echo data of High Resolution Spaceborne SAR is handled; A kind of formation method that is used for the improvement range migration correction algorithm of High Resolution Spaceborne SAR has been proposed; This method is accurately carried out modeling to the warp rail of High Resolution Spaceborne SAR through new oblique distance model; And the 2-d spectrum of the High Resolution Spaceborne SAR echo of having derived on this basis; Range migration correction, stolt interpolation and the orientation of having improved traditional RMA algorithm are to focusing, and the imaging algorithm after the improvement can be accomplished the imaging processing of High Resolution Spaceborne SAR echo effectively.

The inventive method realizes through following technical proposals:

A kind of formation method that is used for the improvement range migration correction algorithm of High Resolution Spaceborne SAR, the step that this method realizes is following:

Step 1, remaining range migration correction and two-dimension fourier transform

Adopt the method for non-linear ChirpScaling; With echoed signal through distance to FFT and orientation before FFT transforms to two-dimensional frequency; Multiply by the remaining range migration correction factor apart from the Doppler territory to carry out remaining range migration correction earlier, the relative velocity of eliminating satellite and target changes the migration error of bringing with oblique distance;

Wherein, the remaining range migration correction factor is:

Be the range migration correction factor, RCM _ResBe remaining range migration amount, λ is a signal wavelength, and Δ τ is a target and with reference to the pairing mistiming of the range difference between the oblique distance, and k is a chirp rate, and τ ' is that distance after the remaining range migration correction is to the time; Described remaining range migration amount is:

${\mathrm{RCM}}_{\mathrm{res}}=\frac{\mathrm{\Δ\τ}}{{\mathrm{\α}}_n\left(f_a\right)}+\mathrm{\β}\left(f_a\right){\mathrm{\Δ\τ}}^2---\left(2\right)$

Wherein, r ₀Satellite is to the minimum distance of target, f _aBe SAR echoed signal orientation to frequency, α _n(f _a) be the second order factor of remaining range migration correction, β (f _a) be three subfactors of remaining range migration correction;

After accomplishing remaining range migration correction, echoed signal is transformed to two-dimensional frequency to FFT and orientation to FFT through distance, and accomplish distance to focusing in two-dimensional frequency.

Step 2, improved stolt interpolation

1. set up the oblique distance model

Setting up the oblique distance model is:

$R(t;{\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="DEST\_PATH\_HDA00002044601500011.png"\; state="\{\{state\}\}">r</figure-callout>}}_0,x_0)=\frac{\sqrt{{\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="DEST\_PATH\_HDA00002044601500011.png"\; state="\{\{state\}\}">r</figure-callout>}}_0^2+{\left[v(t-x_0/v)\right]}^2-2r_{0}v\cos \mathrm{\&theta;}}+v^2(t-x_0/v)/c+a\left(r_0\right){(t-x_0/v)}^3+b\left(r_0\right){(t-x_0/v)}^4}{1-v^2/c^2}---\left(3\right)$

Wherein, R (t; r ₀, x ₀) be the distance between satellite and the target, r ₀Satellite is to the minimum distance of target, x ₀Be target in the orientation to coordinate, θ is the angle of squint, c is the light velocity, t be the orientation to the time, v is the relative velocity between satellite and the target, a, b are respectively crooked cubic term and four items of expression satellite orbit.

2. the calculating of 2-d spectrum

Under this oblique distance model, the expression formula of High Resolution Spaceborne SAR echo is:

$(\mathrm{\&tau;},t;x_0,r_0)=\mathrm{rect}\left[\frac{\mathrm{\&tau;}-\frac{2R(t;{\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="DEST\_PATH\_HDA00002044601500011.png"\; state="\{\{state\}\}">r</figure-callout>}}_0,x_0)}{c}}{T_p}\right]\exp \left\{\mathrm{j\&pi;}{k}_r{[\mathrm{\&tau;}-\frac{2R(t;{\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="DEST\_PATH\_HDA00002044601500011.png"\; state="\{\{state\}\}">r</figure-callout>}}_0,x_0)}{c}]}^2\right\}\exp \{-j\frac{4\mathrm{\&pi;}}{\mathrm{\&lambda;}}R(t;{\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="DEST\_PATH\_HDA00002044601500011.png"\; state="\{\{state\}\}">r</figure-callout>}}_0,x_0)\}---\left(4\right)$

Wherein, s (τ, t; x ₀, r ₀) be the echo of High Resolution Spaceborne SAR, the envelope that rect [] representative transmits, τ be distance to the time, k _rBe the chirp rate that transmits, T _pIt is the pulse width that transmits;

Adopt method in the phase bit, with the High Resolution Spaceborne SAR echo carry out distance to FFT and orientation after the FFT conversion, it is following to obtain the High Resolution Spaceborne SAR 2-d spectrum:

$s(f_{\mathrm{\&tau;}},f_a)=\exp \{-\mathrm{j\&pi;}\frac{f_{\mathrm{\&tau;}}^2}{k_r}\}\exp \{-j{r}_0\sqrt{K_r^2-{(K_x+\frac{v}{c}{K}_r)}^2}\}\exp \{-j\frac{4\mathrm{\&pi;}(f_c+f_{\mathrm{\&tau;}})}{c}(a\left(r_0\right)t_s^3+b\left(r_0\right)t_s^4)\}\exp \{-j2\mathrm{\&pi;}{f}_a\frac{x_0}{v}\}---\left(5\right)$

Wherein, S (f _τ, f _a) be the 2-d spectrum of High Resolution Spaceborne SAR echo data, f _τRepresent distance to frequency, K _rRepresent distance to wave number, K _xRepresent the orientation to wave number, f _cThe centre frequency that representative transmits, t _sRepresent the orientation to site in the phasing;

3. improve the stolt interpolation

The stolt interpolation compensating factor of echo data after two-dimensional frequency multiply by improvement with High Resolution Spaceborne SAR carries out interpolation again and accomplishes the stolt interpolation;

Wherein, according to the High Resolution Spaceborne SAR 2-d spectrum, the stolt interpolation compensating factor after being improved is:

Wherein,

Be the phase place of stolt interpolation compensating factor, r _RefIt is reference distance;

Step 3, two-dimentional IFFT and improved orientation focus on

After accomplishing improved stolt interpolation, the echo data of High Resolution Spaceborne SAR is accomplished range migration correction through improved stolt interpolation, carry out then distance to IFFT and orientation to the IFFT conversion; Simultaneously, in the end the orientation was carried out to focus error compensation through the mode that multiply by compensating factor, the image after obtaining focusing in an orientation before IFFT;

Wherein, the compensating factor expression formula is:

$s_{\mathrm{ac}}(\mathrm{\&tau;},f_a,r_0)=\exp \left\{\mathrm{j\&pi;}\frac{{\mathrm{\&lambda;r}}_0{f}_a^2[v^2\left(r_{\mathrm{ref}}\right)-v^2\left(r_0\right)]}{{2v}^2\left(r_0\right)v^2\left(r_{\mathrm{ref}}\right)}\right\}---\left(7\right)$

Wherein, s _Ac(τ, f _a, f ₀) be the orientation to the focus error compensation factor, v (r _Ref) be the relative velocity at reference distance place, v (r ₀) be the relative velocity at target oblique distance place;

Since then, just accomplished a kind of formation method that is used for the improvement range migration correction algorithm of High Resolution Spaceborne SAR.

Beneficial effect

A kind of formation method that is used for the improvement range migration correction algorithm of High Resolution Spaceborne SAR that the present invention proposes, the contrast prior art, the echo-wave imaging that can accomplish effectively under the High Resolution Spaceborne SAR situation is handled, and its effect is specific as follows:

1. the present invention's relative velocity of having overcome satellite and target changes the range migration error of bringing with oblique distance, can accomplish accurate range migration correction;

2. the present invention is based on new oblique distance model and accurate 2-d spectrum, guarantee the track of High Resolution Spaceborne SAR is carried out accurate modeling, can accomplish accurate stolt interpolation;

3. the present invention's relative velocity of eliminating satellite and target changes the orientation brought to phase error with oblique distance, can accomplish accurate orientation to focusing.

Description of drawings

Fig. 1 is for improving algorithm and traditional algorithm focusing effect comparison diagram;

Fig. 2 is the remaining range migration correction synoptic diagram of embodiment of the present invention.

Embodiment

Elaborate below in conjunction with the embodiment of accompanying drawing to the inventive method.

A kind of formation method that is used for the improvement range migration correction algorithm of High Resolution Spaceborne SAR, the process that this method realizes is following:

Step 1, remaining range migration correction and two-dimension fourier transform

Traditional RMA algorithm thinks that the relative velocity v between satellite and the target is constant, and in fact it is along with oblique distance changes.Therefore, under the High Resolution Spaceborne SAR situation, traditional RMA algorithm can be introduced range migration sum of errors orientation to focusing error.

The present invention adopts the method for non-linear ChirpScaling; With echoed signal through distance to FFT and orientation before FFT transforms to two-dimensional frequency; Multiply by the remaining range migration correction factor apart from the Doppler territory to carry out remaining range migration correction earlier, the relative velocity of eliminating satellite and target changes the migration error of bringing with oblique distance.

Wherein, the remaining range migration correction factor is:

Be the range migration correction factor, RCM _ResBe remaining range migration amount, λ is a signal wavelength, and Δ τ is a target and with reference to the pairing mistiming of the range difference between the oblique distance, and k is a chirp rate, and τ ' is that distance after the remaining range migration correction is to the time; Described remaining range migration amount is:

${\mathrm{RCM}}_{\mathrm{res}}=\frac{\mathrm{\&Delta;\&tau;}}{{\mathrm{\&alpha;}}_n\left(f_a\right)}+\mathrm{\&beta;}\left(f_a\right){\mathrm{\&Delta;\&tau;}}^2---\left(9\right)$

Wherein, r ₀Satellite is to the minimum distance of target, f _aBe the synthetic-aperture radar orientation to frequency, α _n(f _a) be the second order factor of remaining range migration correction, β (f _a) be three subfactors of remaining range migration correction;

After accomplishing remaining range migration correction, echoed signal is transformed to two-dimensional frequency to FFT and orientation to FFT through distance, and accomplish distance to focusing in two-dimensional frequency.

The synoptic diagram of remaining range migration correction is as shown in Figure 2, and blue solid lines is the curve before the remaining range migration correction, and red dot-and-dash line is the curve after the remaining range migration correction.Carry out after the remaining range migration correction, just can better carry out the two-dimentional matched filtering of signal.

Step 2, improvement stolt interpolation

Because the curved in tracks characteristic of High Resolution Spaceborne SAR is obvious, therefore need set up new oblique distance model, and the accurate 2-d spectrum of on the basis of accurate model, deriving, to improve the stolt interpolation, make it under high definition case, to keep accurately.Therefore, the present invention improves the stolt interpolation through following three steps.

1. accurate oblique distance Model Calculation

In order accurately to describe the orbital characteristics of High Resolution Spaceborne SAR, traditional satellite-borne SAR echo oblique distance model is no longer suitable, the formula below the high-precision oblique distance model that the present invention provides adopts:

$R(t;{\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="DEST\_PATH\_HDA00002044601500011.png"\; state="\{\{state\}\}">r</figure-callout>}}_0,x_0)=\frac{\sqrt{{\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="DEST\_PATH\_HDA00002044601500011.png"\; state="\{\{state\}\}">r</figure-callout>}}_0^2+{\left[v(t-x_0/v)\right]}^2-2r_{0}v\cos \mathrm{\&theta;}}+v^2(t-x_0/v)/c+a\left(r_0\right){(t-x_0/v)}^3+b\left(r_0\right){(t-x_0/v)}^4}{1-v^2/c^2}---\left(3\right)$

Wherein, R (t; r ₀, x ₀) be the distance between satellite and the target, r ₀Satellite is to the minimum distance of target, x ₀Be target in the orientation to coordinate, θ is the angle of squint, c is the light velocity, t be the orientation to the time, v is the relative velocity between satellite and the target, a, b are respectively crooked cubic term and four items of expression satellite orbit.

In fact radical sign item in the following formula is exactly traditional echo oblique distance model; Because traditional satellite-based track of echo oblique distance model is the hypothesis of straight line; And the track of real satellite is crooked, and under the situation of High Resolution Spaceborne SAR, the approximate error of conventional model can not ignore; The outer item of radical sign that improves in the echo oblique distance model can accurately be described the track of satellite, is accurate echo oblique distance model therefore.

2. accurate 2-d spectrum calculates

Under this oblique distance model, the expression formula of High Resolution Spaceborne SAR echo is:

$(\mathrm{\&tau;},t;x_0,r_0)=\mathrm{rect}\left[\frac{\mathrm{\&tau;}-\frac{2R(t;{\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="DEST\_PATH\_HDA00002044601500011.png"\; state="\{\{state\}\}">r</figure-callout>}}_0,x_0)}{c}}{T_p}\right]\exp \left\{\mathrm{j\&pi;}{k}_r{[\mathrm{\&tau;}-\frac{2R(t;{\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="DEST\_PATH\_HDA00002044601500011.png"\; state="\{\{state\}\}">r</figure-callout>}}_0,x_0)}{c}]}^2\right\}\exp \{-j\frac{4\mathrm{\&pi;}}{\mathrm{\&lambda;}}R(t;{\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="DEST\_PATH\_HDA00002044601500011.png"\; state="\{\{state\}\}">r</figure-callout>}}_0,x_0)\}---\left(4\right)$

Wherein, s (τ, t; x ₀, r ₀) be the echo of High Resolution Spaceborne SAR, the envelope that rect [] representative transmits, τ be distance to the time, k _rBe the chirp rate that transmits, T _pBe the pulse width that transmits, λ is a signal wavelength.

High-precision stolt interpolation needs high-precision 2-d spectrum, therefore in the calculating of the above-mentioned enterprising capable 2-d spectrum in accurate oblique distance model basis.Utilize method in the phase bit, with above-mentioned High Resolution Spaceborne SAR echo carry out distance to FFT and orientation after FFT, can obtain 2-d spectrum and be:

$s(f_{\mathrm{\&tau;}},f_a)=\exp \{-\mathrm{j\&pi;}\frac{f_{\mathrm{\&tau;}}^2}{k_r}\}\exp \{-j{r}_0\sqrt{K_r^2-{(K_x+\frac{v}{c}{K}_r)}^2}\}\exp \{-j\frac{4\mathrm{\&pi;}(f_c+f_{\mathrm{\&tau;}})}{c}(a\left(r_0\right)t_s^3+b\left(r_0\right)t_s^4)\}\exp \{-j2\mathrm{\&pi;}{f}_a\frac{x_0}{v}\}---\left(5\right)$

Wherein, S (f _τ, f _a) be the 2-d spectrum of High Resolution Spaceborne SAR echo data, f _τRepresent distance to frequency, K _rRepresent distance to wave number, K _xRepresent the orientation to wave number, f _cThe centre frequency that representative transmits, t _sRepresent the orientation to site in the phasing.

3. improve the stolt interpolation

Through to the accurate oblique distance model of High Resolution Spaceborne SAR with after 2-d spectrum calculates accurately, can further improve to obtain good imaging effect the stolt interpolation of traditional RMA algorithm.

Traditional algorithm multiply by stolt interpolation compensating factor (second conjugation in the formula (5)) in two-dimensional frequency earlier, carries out interpolation again and accomplishes the stolt interpolation.Formula (5) is compared with traditional frequency spectrum and has been had more about t _sCubic term and four items, promptly the 3rd, this also need compensate.

Therefore, to the High Resolution Spaceborne SAR system, wherein, according to the High Resolution Spaceborne SAR 2-d spectrum, the stolt interpolation compensating factor after being improved is:

Wherein,

Be the phase place of stolt interpolation compensating factor, r _RefIt is reference distance;

Step 3, two-dimentional IFFT and improved orientation focus on

Accomplish after the improved stolt interpolation, the echo data of High Resolution Spaceborne SAR just can be accomplished accurate range migration correction, then carry out distance to IFFT and orientation to the IFFT conversion.

Then, the relative velocity of satellite above-mentioned and target also need improve in the RMA algorithm to focusing error with the orientation that the oblique distance variation brings.The present invention carried out this focus error compensation in last orientation of traditional RMA algorithm before IFFT, the compensating factor expression formula is:

$s_{\mathrm{ac}}(\mathrm{\&tau;},f_a,r_0)=\exp \left\{\mathrm{j\&pi;}\frac{{\mathrm{\&lambda;r}}_0{f}_a^2[v^2\left(r_{\mathrm{ref}}\right)-v^2\left(r_0\right)]}{{2v}^2\left(r_0\right)v^2\left(r_{\mathrm{ref}}\right)}\right\}---\left(7\right)$

Wherein, s _Ac(τ, f _a, r ₀) be the orientation to the focus error compensation factor, v (r _Ref) be the relative velocity at reference distance place, v (r ₀) be the relative velocity at target oblique distance place.

Accomplish improved orientation after focusing on, the echo data of High Resolution Spaceborne SAR has just been accomplished accurate orientation to focusing, the image after also just having obtained focusing on.

Since then, just accomplished a kind of formation method that is used for the improvement range migration correction algorithm of High Resolution Spaceborne SAR.

Embodiment

For the improvement RMA algorithm of verifying that the present invention provides, carried out the echo simulation of high resolving power (0.15m) satellite-borne SAR, the parameter of emulation is as shown in table 1.

And the RMA algorithm after using traditional RMA algorithm respectively and improving carries out imaging processing, and the result that obtains is as shown in Figure 1.Can obviously be observed out by figure, the focusing outcome quality of traditional RMA algorithm is very poor, and the orientation is to secondary lobe broadening and bending, and improved RMA algorithm focusing result is good.Therefore can prove that improved RMA algorithm can carry out good imaging processing to the High Resolution Spaceborne SAR echo data effectively.

Claims (1)

1. formation method that is used for the improvement range migration correction algorithm of High Resolution Spaceborne SAR, the step that this method realizes is following:

Step 1, remaining range migration correction and two-dimension fourier transform

Adopt the method for non-linear ChirpScaling; With echoed signal through distance to FFT and orientation before FFT transforms to two-dimensional frequency; Multiply by the remaining range migration correction factor apart from the Doppler territory to carry out remaining range migration correction earlier, the relative velocity of eliminating satellite and target changes the migration error of bringing with oblique distance;

Wherein, the remaining range migration correction factor is:

Be the range migration correction factor, RCM _ResBe remaining range migration amount, λ is a signal wavelength, and Δ τ is a target and with reference to the pairing mistiming of the range difference between the oblique distance, and k is a chirp rate, and τ ' is that distance after the remaining range migration correction is to the time; Described remaining range migration amount is:

${\mathrm{RCM}}_{\mathrm{res}}=\frac{\mathrm{\&Delta;\&tau;}}{{\mathrm{\&alpha;}}_n\left(f_a\right)}+\mathrm{\&beta;}\left(f_a\right){\mathrm{\&Delta;\&tau;}}^2---\left(2\right)$

Wherein, r ₀Satellite is to the minimum distance of target, f _aBe SAR echo data orientation to frequency, α _n(f _a) be the second order factor of remaining range migration correction, β (f _a) be remaining range migration correction three subfactors;

After accomplishing remaining range migration correction, echoed signal is transformed to two-dimensional frequency to FFT and orientation to FFT through distance, and accomplish distance to focusing in two-dimensional frequency;

Step 2, improved stolt interpolation

1. set up the oblique distance model

Setting up the oblique distance model is:

$R(t;r_0,x_0)=\frac{\sqrt{r_0^2+{\left[v(t-x_0/v)\right]}^2-2r_{0}v\cos \mathrm{\&theta;}}+v^2(t-x_0/v)/c+a\left(r_0\right){(t-x_0/v)}^3+b\left(r_0\right){(t-x_0/v)}^4}{1-v^2/c^2}---\left(3\right)$

Wherein, R (t; r ₀, x ₀) be the distance between satellite and the target, r ₀Satellite is to the minimum distance of target, x ₀Be target in the orientation to coordinate, θ is the angle of squint, c is the light velocity, t be the orientation to the time, v is the relative velocity between satellite and the target, a, b are respectively crooked cubic term and four items of expression satellite orbit;

2. the calculating of 2-d spectrum

Under this oblique distance model, the expression formula of High Resolution Spaceborne SAR echo is:

$(\mathrm{\&tau;},t;x_0,r_0)=\mathrm{rect}\left[\frac{\mathrm{\&tau;}-\frac{2R(t;r_0,x_0)}{c}}{T_p}\right]\exp \left\{\mathrm{j\&pi;}{k}_r{[\mathrm{\&tau;}-\frac{2R(t;r_0,x_0)}{c}]}^2\right\}\exp \{-j\frac{4\mathrm{\&pi;}}{\mathrm{\&lambda;}}R(t;r_0,x_0)\}---\left(4\right)$

Wherein, s (τ, t; x ₀, r ₀) be the echo of High Resolution Spaceborne SAR, the envelope that rect [] representative transmits, τ be distance to the time, k _rBe the chirp rate that transmits, T _pIt is the pulse width that transmits;

Adopt method in the phase bit, with the High Resolution Spaceborne SAR echo carry out distance to FFT and orientation after the FFT conversion, it is following to obtain the High Resolution Spaceborne SAR 2-d spectrum:

$s(f_{\mathrm{\&tau;}},f_a)=\exp \{-\mathrm{j\&pi;}\frac{f_{\mathrm{\&tau;}}^2}{k_r}\}\exp \{-j{r}_0\sqrt{K_r^2-{(K_x+\frac{v}{c}{K}_r)}^2}\}\exp \{-j\frac{4\mathrm{\&pi;}(f_c+f_{\mathrm{\&tau;}})}{c}(a\left(r_0\right)t_s^3+b\left(r_0\right)t_s^4)\}\exp \{-j2\mathrm{\&pi;}{f}_a\frac{x_0}{v}\}---\left(5\right)$

Wherein, S (f _τ, f _a) be the 2-d spectrum of High Resolution Spaceborne SAR echo data, f _τRepresent distance to frequency, K _rRepresent distance to wave number, K _xRepresent the orientation to wave number, f _cThe centre frequency that representative transmits, t _sRepresent the orientation to site in the phasing;

3. improve the stolt interpolation

The stolt interpolation compensating factor of echo data after two-dimensional frequency multiply by improvement with High Resolution Spaceborne SAR carries out interpolation again and accomplishes the stolt interpolation;

Wherein, according to the High Resolution Spaceborne SAR 2-d spectrum, the stolt interpolation compensating factor after being improved is:

Wherein,

Be the phase place of stolt interpolation compensating factor, r _RefIt is reference distance;

Step 3, two-dimentional IFFT and improved orientation focus on

After accomplishing improved stolt interpolation, the echo data of High Resolution Spaceborne SAR is accomplished range migration correction through improved stolt interpolation, carry out then distance to IFFT and orientation to the IFFT conversion; Simultaneously, in the end the orientation was carried out to focus error compensation through the mode that multiply by compensating factor, the image after obtaining focusing in an orientation before IFFT;

Wherein, the compensating factor expression formula is:

$s_{\mathrm{ac}}(\mathrm{\&tau;},f_a,r_0)=\exp \left\{\mathrm{j\&pi;}\frac{{\mathrm{\&lambda;r}}_0{f}_a^2[v^2\left(r_{\mathrm{ref}}\right)-v^2\left(r_0\right)]}{{2v}^2\left(r_0\right)v^2\left(r_{\mathrm{ref}}\right)}\right\}---\left(7\right)$

Wherein, s _Ac(τ, f _a, r ₀) be the orientation to the focus error compensation factor, v (r _Ref) be the relative velocity at reference distance place, v (r ₀) be the relative velocity at target oblique distance place;

Since then, just accomplished a kind of formation method that is used for the improvement range migration correction algorithm of High Resolution Spaceborne SAR.

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